Antidifferentiation: The Indefinite
Intergral
Chapter Five
§5.1
Antidifferetiation
§5.1 General Antiderivative of a
Function
§5.1 General Antiderivative of a
Function
§5.1 Rules for Integrating
Common Function
The Constant Rule
§5.1 Rules for Integrating Common
Function
Example:
Solution:
§5.1 Applied Initial Value Problems
A Differential equation is an equation that involves differentials
or derivatives.
An initial Value problems is a problem that involves solving
a differential equation subject to a specified initial condition.
For instance, we were required to find y=f(x) so that
We solved this initial problem by finding the antiderivative
And using the initial condition to evaluate C.
Example:
The population p(t) of a bacterial colony t hours after observation
begins is found to be change at the rate
If the population was 2000,000 bacteria when observations
began, what will be population 12 hours later?
Solution:
§5.2 Integration by Substitution
How to do the following integral?
§5.2 Integration by Substitution
Think of u=u(x) as a change of variable whose differential is
Then
Example:
Find
Solution:
Example:
Solution:
Example:
Solution:
To be continued
Example:
Solution:
Example:
Solution:
§5.3 The Definite Integral and the
Fundamental Theorem of Calculus
All rectangles
have same width.
• n subintervals:
• Subinterval width
•Formula for xi:
• Choice of n evaluation points
Right-endpoint approximation
left-endpoint approximation
Midpoint Approximation
Example:
=0.285
To be continued
=0.3325
=0.385
Example:
left-endpoint approximation
Midpoint Approximation
Right-endpoint approximation
S200=1.098608585
S400 =1.098611363
Area Under a Curve Let f(x) be continuous and satisfy f(x)≥0
on the interval a≤x≤b. Then the region under the curve y=f(x)
over the interval a≤x≤b has area
n
A  lim Sn  lim[ f ( x1 )  f ( x2 )  ...  f ( xn )]x  lim  f ( x j )x
n
n
n
j 1
Where xj is the point chosen from the jth subinterval if the
Interval a≤x≤b is divided into n equal parts, each of length
x 
ba
n
§5.3 The Definite Integral
Riemann sum
Let f(x) be a function that is continuous on
the interval a≤x≤b. Subdivide the interval a≤x≤b into n equal
parts, each of width x  b  a ,and choose a number xk from the
n
kth subinterval for k=1, 2, …, . Form the sum
Called a Riemann sum.
Note: f(x)≥0 is not required
§5.3 The Definite Integral
The Definite Integral
a≤x≤b, denoted by
n→+∞; that is

b
a
the definite integral of f on the interval
, is the limit of the Riemann sum as
f(x)dx
The function f(x) is called the integrand, and the numbers a and
b are called the lower and upper limits of integration,
respectively. The process of finding a definite integral is called
definite integration.
Note: if f(x) is continuous on a≤x≤b, the limit used to define
b
integral a f(x)dx exist and is same regardless of how the subinterval
representatives xk are chosen.
§5.3 Area as Definite Integral
If f(x) is continuous and f(x)≥0 for all
x in [a,b],then b

a
f ( x)dx  0
and equals the area of the region
bounded by the graph f and the x-axis
between x=a and x=b
If f(x) is continuous and f(x)≤0 for all
b
x in [a,b],then
b

a
f ( x)dx  0
And  f ( x)dx equals the area of the
a
region bounded by the graph f and the
x-axis between x=a and x=b
§5.3 Area as Definite Integral

b
a
f ( x)dx equals the difference between the area under the graph
of f above the x-axis and the area above the graph of f below the
x-axis between x=a and x=b
This is the net area of the region bounded by the graph of f and
the x-axis between x=a and x=b
§5.3 The Fundamental Theorem of
Calculus
The Fundamental Theorem of Calculus
continuous on the interval a≤x≤b, then

b
a
If the function f(x) is
f ( x)dx  F (b)  F (a)
Where F(x) is any antiderivative of f(x) on a≤x≤b
Another notation:

b
a
f ( x)dx  F ( x) |ba  F (b)  F (a)
§5.3 The Fundamental Theorem of
Calculus (Area justification )
b
In the case of f(X)≥0, a f ( x)dx represents the area the curve
y=f(x) over the interval [a,b]. For fixed x between a and b let A(x)
denote the area under y=f(x) over the interval [a,x].
By the definition of the derivative,
Example
Differentiation
Indefinite Integration
Definite integration
§5.3 Integration Rule
Subdivision
Rule
§5.3
Subdivision Rule
Example
Solution:
Example
Solution:
To be continued
§5.3 Substituting in a definite integral
2.
x2
1 1
2
2 3
dx  
du 
u
x 1
3
3
3
u
x3  1


2
0
x
2
2
2 3
4
dx 
x 1 
3
3
3
x 1
0
§5.3 Substituting in a definite
integral
Example
Solution:
Example
Solution: